Homotopy Analysis Method for Three Types of Fractional Partial Differential Equations

Haidong Qu,Xuan Liu,Zihang She

doi:10.1155/2020/7232907

Abstract

In this paper, three types of fractional order partial differential equations, including the fractional Cauchy–Riemann equation, fractional acoustic wave equation, and two-dimensional space partial differential equation with time-fractional-order, are considered, and these models are obtained from the standard equations by replacing an integer-order derivative with a fractional-order derivative in Caputo sense. Firstly, we discuss the fractional integral and differential properties of several functions which are derived from the Mittag-Leffler function. Secondly, by using the homotopy analysis method, the exact solutions for fractional order models mentioned above with suitable initial boundary conditions are obtained. Finally, we draw the computer graphics of the exact solutions, the approximate solutions (truncation of finite terms), and absolute errors in the limited area, which show that the effectiveness of the homotopy analysis method for solving fractional order partial differential equations.

Highlights

Many effective methods for fractional differential equations have been presented, such as the finite difference method [9], spectral method [5], matrix approach [10], homotopy analysis method (HAM) [11], and homotopy perturbation method [12]
In 2007, firstly, the HAM that was developed for an integer-order differential equation was directly extended to derive explicit and numerical solutions of nonlinear fractional differential equations by by Song and Zhang [19]
In 2008, Xu and Cang [20] employed the HAM to derive the solutions of the time fractional wave-like differential equations with a variable coefficient

Summary

Definitions and Lemmas

Definition 1. (see [3]). A real function f(x), x > 0, is said to be in the space Cμ, μ ∈ R, if there exists a real number p > μ, such that f(x) xpf1(x), where f1(x) ∈ C(0,∞), and it is said to be in the space Cnμ, if and only if fn ∈ Cμ, n ∈ N. E Riemann–Liouville fractional integral of order α ∈ R, α > 0 of a function f(x) ∈ Cμ, μ ≥ − 1 is defined as. We denote the beta function by β􏽥, and according to the definition of the Riemann–Liouville fractional integral, we have. While replacing the Riemann–Liouville fractional derivative with the Caputo derivative, we get the following forms: cDαa+sinα􏼂λ(t − a)α􏼃􏼁(x) λcosα􏼂λ(x − a)α􏼃, cDαa+cosα􏼂λ(t − a)α􏼃􏼁(x) −λsinα􏼂λ(x − a)α􏼃. While, replacing the Riemann–Liouville fractional derivative with the Caputo derivative, we get the following forms: cDαa+sinhα􏼂λ(t − a)α􏼃􏼁(x) λcoshα􏼂λ(x − a)α􏼃, cDαa+coshα􏼂λ(t − a)α􏼃􏼁(x) λsinhα􏼂λ(x − a)α􏼃. Let u0(x, t) denote an initial approximation of the solution of equation (31), h a nonzero auxiliary parameter, H(x, t) a nonzero auxiliary function, and L is an auxiliary linear operator. It is easy to obtain u1(x, t), u2(x, t), . . . one after another, and we get an exact solution of the original equation (31) of the series form: u(x, t) 􏽘 um(x, t)

Applying HAM

Conclusions